Momentum-Space Josephson Effects

The Josephson effect is a prominent phenomenon of quantum supercurrents that has been widely studied in superconductors and superfluids. Typical Josephson junctions consist of two real-space superconductors (superfluids) coupled through a weak tunneling barrier. Here we propose a momentum-space Josephson junction in a spin-orbit coupled Bose-Einstein condensate, where states with two different momenta are coupled through Raman-assisted tunneling. We show that Josephson currents can be induced not only by applying the equivalent of “voltages,” but also by tuning tunneling phases. Such tunneling-phase-driven Josephson junctions in momentum space are characterized through both full mean field analysis and a concise two-level model, demonstrating the important role of interactions between atoms. Our scheme provides a platform for experimentally realizing momentum-space Josephson junctions and exploring their applications in quantum-mechanical circuits.

Figure 1

(a) Illustration of a conventional JJ for real-space superconductors (top) versus MSJJ (bottom), where the double-well band dispersion is generated using a spin-orbit coupled BEC. (b),(c) Experimental setup for realizing a MSJJ. Two pairs of Raman lasers realize SOC (blue) and weak coupling (red) between two band minima, respectively.

Figure 2

(a) Ground state phase diagram, where $\mathrm{\Omega }=2.7$, ${\varphi }_L=0$ and $ng=0.07$. The inset shows the first-order phase transition for small ${\mathrm{\Omega }}_L$. Black, dark gray, light gray, and white lines correspond to ${\mathrm{\Omega }}_L=0.01$, 0.1, 0.2, and 0.5, respectively. (b) Real space density modulation for the ground state with parameters $\delta =0.054$ and ${\mathrm{\Omega }}_L=0.015$ as denoted by the black cross in (a). (c) Illustration of induced couplings between six most relevant momentum states.

Figure 3

(a) Time evolution of the populations at different momentum states for a tunneling-phase-driven MSJJ for $\mathrm{\Omega }=2.7$, $\delta =0.014$, ${\mathrm{\Omega }}_L=0.015$, and $ng=0.07$. Evolutions of phase difference (b) and polarization (c) for Josephson oscillation (solid blue) and plasma oscillation (dashed orange). (d) Oscillation period $T$ versus $\delta$ when ${\varphi }_L$ is quenched from ${\varphi }_{L0}$ to 0 at $\mathrm{\Omega }=2.7$. Circles are results from the GPE simulation, while solid ($ng=0.07$) and dashed (single particle) lines are from the two-level model. Different colors correspond to different parameter sets: ${\mathrm{\Omega }}_L=0.015$, ${\varphi }_{L0}=0.4\pi$ (blue); ${\mathrm{\Omega }}_L=0.025$, ${\varphi }_{L0}=0.4\pi$ (orange); and ${\mathrm{\Omega }}_L=0.015$, ${\varphi }_{L0}=0.2\pi$ (red). Blue and red dashed lines overlap (purple) since $T$ is independent of ${\varphi }_{L0}$ for the single particle case.

Figure 4

(a) Classical trajectories in the $z\text{−}\varphi$ plane for $0<{\varphi }_{L0}\le \pi$, with initial value of $z$ at 0.434. (b) Same as (a) but generated through the GPE simulation. Parameters are $ng=0.07$, $\delta =0.008$ (corresponding to initial polarization 0.434), $\mathrm{\Omega }=2.7$, and ${\mathrm{\Omega }}_L=0.015$. The three colors correspond to ${\varphi }_{L0}=0.2$ (blue), 0.4 (orange), 0.8 (green), respectively. The arrows denote the direction of each trajectory.

Figure 5

(a) Self-trapping effects from the GPE simulation. The curves correspond to $ng=0.07$ (blue), 0.14 (orange), and 0.35 (green), for $\mathrm{\Omega }=2.7$, ${\mathrm{\Omega }}_L=0.03$, ${\varphi }_{L0}=0.2\pi$, and $\delta =0$. (b) Classical phase plane from the two-level model. The colors are the same as in (a) except $ng=0.1$ for the red curve.